This presentation laws of absorption of light- lambert's law and beer's law and their mathematical derivation

1. Dr. Y. S. THAKARE
M.Sc. (CHE) Ph D, NET, SET
Assistant Professor in Chemistry,
Shri Shivaji Science College, Amravati
Email: yogitathakare_2007@rediffmail.com
B Sc- III Year
SEM-V
PAPER-III
PHYSICAL CHEMISTRY
UNIT- V
Laws of Absorption of light
Lambert’s- Beer’s Law
29-August -20 1

2. Laws of Absorption of Light
The portion of the light absorbed by a medium is governed by two laws.
i) Lambert's law ii) Beer's law
i) Lambert's law: In 1758, J.H. Lambert derived a relationship
between the amount of light absorbed and the thickness or depth
of the absorbing material. According to this law, when a beam of
monochromatic light passes through a homogeneous absorbing
medium, the rate of decrease of the intensity of the radiation with
thickness of absorbing medium is proportional to the intensity of
the incident radiation.
I=I0 I=I
x=0 x=x
29-August -20 Dr. Yogita Sahebrao Thakare

3. Mathematically this law can be expressed as
−
𝑑𝐼
𝑑𝑥
∝ Ι
−
𝑑𝐼
𝑑𝑥
= 𝑘Ι ……….(1)
Where, I = Intensity of incident light.
x = Thickness of the medium,
k = Constant called absorption coefficient.
On rearranging Eq. 1 we get
𝑑𝐼
Ι
= −k 𝑑𝑥 ……….(2)
Integrating Eq. 2 between the limits. I = Io at x = 0 and I = I at x = x, we get
𝐼=𝐼0
𝐼=𝐼 𝑑𝐼
Ι
= −k 𝑥=0
𝑥=𝑥
𝑑𝑥 ……….(3)
or ln
Ι
Io
= −k𝑥 ……….(4)
I/Io = e-kx
I=Ioe-kx ……….(5)
According to Eq. 5, the intensity of beam of monochromatic radiation decreases
exponentially with increase in the thickness 'x' of the absorbing substance. This is
Lamberts law.29-August -20 Dr. Yogita Sahebrao Thakare

4. Eq. 4 can be written as
2.303 log
Ι
Io
= - k x ……….(7)
log
Ι
Io
= −
k
2.303
x ……….(8)
log
Ι
Io
= - ∈ x ……….(9)
log
Ι0
I
= ∈ x ……….(10)
Where ∈ =
k
2.303
and is called extinction coefficient. It depends upon nature of
the absorbing substance and wavelength of incident light.
29-August -20 Dr. Yogita Sahebrao Thakare

5. ii) Beer's law : Beer (1852) extended the Lambert's law to solution. This law when
applied to solutions is called as Beers-Lambert's law.
Statement : When a beam of monochromatic light passes through a solution of an
absorbing substance, the rate of decrease of intensity of light with thickness of
solution is directly proportional to the intensity of incident light as well as
concentration of the solution.
I=I0 I=I
x=0 x=x
29-August -20 Dr. Yogita Sahebrao Thakare

6. Mathematical Expression:
Mathematically, the law can be expressed as
−
𝑑𝐼
𝑑𝑥
∝ I × C
−
𝑑𝐼
𝑑𝑥
= 𝛼 I. C ……….(1)
Where, I = Intensity of incident light.
x = Thickness of the medium.
𝛼 = Constant called absorption coefficient.
On rearranging eq. 1 we get
𝑑𝐼
Ι
= −𝛼 𝐶 𝑑𝑥 ……….(2)
Integrating eq.2 between the limit I = Io at x = 0 and I = I at x = x, we get
𝐼=𝐼0
𝐼=𝐼 𝑑𝐼
Ι
= −𝛼 𝐶 𝑥=0
𝑥=𝑥
𝑑𝑥 ……….(3)
or ln
Ι
Io
= −𝛼 𝐶𝑥 ……….(4)
or I/Io = 𝑒−𝛼𝐶𝑥
……….(5)
I=Io 𝑒−𝛼𝐶𝑥
……….(6)
According to eq.6, the intensity of beam of monochromatic radiation decreases exponentially
with increase in the thickness X and the concentration 'C of the absorbing substance. This is
mathematical expression of Beer's law.
29-August -20 Dr. Yogita Sahebrao Thakare

7. Eq. 4 can be written as
2.303 log
Ι
Io
= - 𝛼 C x ……….(7)
or log I/Io = −
𝛼
2.303
C x ……….(8)
or log
Io
I
=
𝛼
2.303
C x ……….(9)
or log
Io
I
=∈ C x ……….(10)
Where,
𝛼
2.303
= ∈ and is called molar absorption coefficient or molar
absorptivity or molar extinction coefficient of the absorbing medium. It is
characteristic of the solute and depends upon nature of solvent, temperature and
wavelength of light employed. If concentration 'C' is expressed in mol dm-3 and path
length ‘x’ in cm then ∈ (dm3 mol-1 cm-1) is referred to as molar absorption coefficient
or molar extinction coefficient
The quantity log
Io
I
is called optical density or absorbance of the medium. It is
denoted by A.
log
Io
I
= 𝐴 =∈ C x ……….(11)
29-August -20 Dr. Yogita Sahebrao Thakare

8. log
Io
I
= 𝐴 =∈ C x
when C=1 mole lit-1 , x=1cm then ∈ = A
Defination - Molar absorption coefficient or molar absorptivity or molar
may be defined, as the absorbance for a concentration of 1 mole per liter of
substance and optical path length of 1 cm.
The transmittance T is defined as
𝑇 =
I
I0
OR A= log
Io
I
= log
1
I I0
= log
1
T
= 𝑙𝑜𝑔𝑇−1 = −𝑙𝑜𝑔𝑇
29-August -20 Dr. Yogita Sahebrao Thakare